Simplicial matrices and the nerves of weak n-categories I:
nerves of bicategories

John W. Duskin

To a bicategory B
(in the sense of Bénabou) we
assign a simplicial set Ner(B), the (geometric) nerve of
B, which completely encodes the structure of B
as a bicategory. As a simplicial set Ner(B) is a subcomplex of its
2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we
call a 2-dimensional Postnikov complex. We then give,
somewhat more delicately, a complete characterization of those
simplicial sets which are the nerves of bicategories as certain
2-dimensional Postnikov complexes which satisfy certain restricted
`exact horn-lifting' conditions whose satisfaction is controlled by
(and here defines) subsets of (abstractly) invertible 2 and
1-simplices. Those complexes which have, at minimum, their
degenerate 2-simplices always invertible and have an invertible
2-simplex $\chi_2^1(x_{12}, x_{01})$ present for each `composable
pair' $(x_{12}, \_ , x_{01}) \in \mhorn_2^1$ are exactly the
nerves of bicategories. At the other extreme, where all 2
and 1-simplices are invertible, are those Kan complexes in which
the Kan conditions are satisfied exactly in all dimensions >2.
These are exactly the nerves of bigroupoids - all 2-cells are
isomorphisms and all 1-cells are equivalences.